0,2

A. Cayley, A memoir on the transformation of elliptic functions, Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 9, p. 128.

H. R. P. Ferguson, D. E. Nielsen and G. Cook, A partition formula for the integer coefficients of the theta function nome, Math. Comp., 29 (1975), 851-855.

N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; Eq. (34.3).

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n=0..1000

G.f.: Product ( 1 - x^k )^(-c(k)); c(k) = 2, 2, 2, 0, 2, 2, 2, 0, ....

G.f.: eta(q^4)^2/(eta(q)^2*q^(1/4)) where eta = Dedekind's function.

A079006(n) = (-1)^n a(n).

Expansion of q^(-1/4)(1/2)(k/k')^(1/2) in powers of q.

Euler transform of period 4 sequence [2, 2, 2, 0, ...].

Given g.f. A(x), then B(x)=(x*A(x^4))^4 satisfies 0=f(B(x), B(x^2)) where f(u, v)=(1+16u)(1+16v)v-u^2 . - Michael Somos Jul 09 2005

Given g.f. A(x), then B(x)=x*A(x^4) satisfies 0=f(B(x), B(x^3)) where f(u, v)=(u^2+v^2)^2 -uv(1+4uv)^2 . - Michael Somos Jul 09 2005

G.f.: (Product_{k>0} (1+x^(2k))/(1-x^(2k-1)))^2 = (Product_{k>0} (1-x^(4k))/(1-x^k))^2 = square of g.f. for A001935.

with (numtheory): etr:= proc(p) local b; b:=proc(n) option remember; local d, j; if n=0 then 1 else add (add (d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n fi end end: a:= etr (n-> [2, 2, 2, 0] [modp(n-1, 4)+1]): seq (a(n), n=0..38); [From Alois P. Heinz (heinz(AT)hs-heilbronn.de), Sep 08 2008]

(PARI) a(n)=if(n<0, 0, polcoeff((eta(x^4+x*O(x^n))/eta(x+x*O(x^n)))^2, n))

(PARI) a(n)=if(n<0, 0, polcoeff(prod(k=1, n, 1/if(k%4, 1-x^k, 1), 1+x*O(x^n))^2, n))

See A127391, A127392, A079006 for other versions of this sequence.

A079006(n)=(-1)^n a(n). Convolution square of A001935.

Sequence in context: A006327 A103577 A079006 this_sequence A127297 A018739 A011893

Adjacent sequences: A001933 A001934 A001935 this_sequence A001937 A001938 A001939

nonn

N. J. A. Sloane (njas(AT)research.att.com).